Theorem nfcnv 5885

Description: Bound-variable hypothesis builder for converse relation. (Contributed by NM, 31-Jan-2004.) (Revised by Mario Carneiro, 15-Oct-2016.)

Hypothesis

Ref	Expression
nfcnv.1	⊢ Ⅎ𝑥𝐴

Assertion

Ref	Expression
nfcnv	⊢ Ⅎ𝑥◡𝐴

Proof of Theorem nfcnv

Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-cnv 5690	. 2 ⊢ ◡𝐴 = {⟨𝑦, 𝑧⟩ ∣ 𝑧𝐴𝑦}
2		nfcv 2899	. . . 4 ⊢ Ⅎ𝑥𝑧
3		nfcnv.1	. . . 4 ⊢ Ⅎ𝑥𝐴
4		nfcv 2899	. . . 4 ⊢ Ⅎ𝑥𝑦
5	2, 3, 4	nfbr 5199	. . 3 ⊢ Ⅎ𝑥 𝑧𝐴𝑦
6	5	nfopab 5221	. 2 ⊢ Ⅎ𝑥{⟨𝑦, 𝑧⟩ ∣ 𝑧𝐴𝑦}
7	1, 6	nfcxfr 2897	1 ⊢ Ⅎ𝑥◡𝐴

Syntax hints:  Ⅎwnfc 2879   class class class wbr 5152  {copab 5214  ^◡ccnv 5681

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-br 5153  df-opab 5215  df-cnv 5690

This theorem is referenced by:  nfrn  5958  nfpred  6315  nffun  6581  nff1  6796  nfsup  9482  nfinf  9513  gsumcom2  19937  ptbasfi  23505  mbfposr  25601  itg1climres  25664  funcnvmpt  32474  nfwsuc  35447  aomclem8  42516  rfcnpre1  44412  rfcnpre2  44424  smfpimcc  46225